It is well known that the medians of a triangle divide it into 6 triangles.

It is less known that the centers of their circumscribed circles belong to one circumference as drawn below

This remarkable theorem was proved in the 21st century! Unfortunately, I lost its source.

I can't prove this difficult theorem by hand. However, I can prove it with Maple.

The aim of this post is to expose these proofs. Everybody knows that it is scarcely possible

to construct a general triangle with help of the geometry package of Maple.

Without loss of generality one may assume that the vertex A is placed at the origin,

the vertex B is placed at (1,0), and the vertex C(xC,yC). We firstly consider the theorem

in the case of concrete values of xC and yC.

restart; with(geometry):with(plots):

point(A, 0, 0);

point(B, 1, 0);

xC := 15*(1/10); yC := sqrt(3); point(C, xC, yC);

triangle(T, [A, B, C]);

median(mA, A, T, MA);

median(mB, B, T, MB);

median(mC, C, T, MC);

line(m1, [A, MA]);

line(m2, [B, MB]);

intersection(E, m1, m2);

triangle(AEMB, [A, E, MB]);

circumcircle(c1, AEMB, 'centername' = C1);

circumcircle(c2, triangle(CEMB, [C, E, MB]), 'centername' = C2);

circumcircle(c3, triangle(CEMA, [C, E, MA]), 'centername' = C3);

circumcircle(c4, triangle(BEMA, [B, E, MA]), 'centername' = C4);

circumcircle(c5, triangle(BEMC, [B, E, MC]), 'centername' = C5);

circumcircle(c6, triangle(AEMC, [A, E, MC]), 'centername' = C6);

circle(CC, [C1, C2, C3]);

IsOnCircle(C4, CC);

true

IsOnCircle(C5, CC);

true

IsOnCircle(C6, CC);

true

display([draw([T(color = black), mA(color = black), mB(color = black), mC(color = black), C1(color = blue), C2(color = blue), C3(color = blue), C4(color = blue), C5(color = blue), C6(color = blue), CC(color = red)], symbol = solidcircle, symbolsize = 15, thickness = 2, scaling = constrained), textplot({[-0.5e-1, 0.5e-1, "A"], [.95, 0.5e-1, "B"], [xC-0.5e-1, yC+0.5e-1, "C"]})], axes = frame, view = [-.1 .. max(1, xC)+.1, 0 .. yC+.1]);

This can be done as a procedure in such a way.

restart; SixPoints := proc (xC, yC) geometry:-point(A, 0, 0); geometry:-point(B, 1, 0); geometry:-point(C, xC, yC); geometry:-triangle(T, [A, B, C]); geometry:-median(mA, A, T, MA); geometry:-median(mB, B, T, MB); geometry:-median(mC, C, T, MC); geometry:-line(m1, [A, MA]); geometry:-line(m2, [B, MB]); geometry:-intersection(E, m1, m2); geometry:-triangle(AEMB, [A, E, MB]); geometry:-circumcircle(c1, AEMB, 'centername' = C1); geometry:-circumcircle(c2, geometry:-triangle(CEMB, [C, E, MB]), 'centername' = C2); geometry:-circumcircle(c3, geometry:-triangle(CEMA, [C, E, MA]), 'centername' = C3); geometry:-circumcircle(c4, geometry:-triangle(BEMA, [B, E, MA]), 'centername' = C4); geometry:-circumcircle(c5, geometry:-triangle(BEMC, [B, E, MC]), 'centername' = C5); geometry:-circumcircle(c6, geometry:-triangle(AEMC, [A, E, MC]), 'centername' = C6); geometry:-circle(CC, [C1, C2, C3]); return geometry:-IsOnCircle(C4, CC), geometry:-IsOnCircle(C5, CC), geometry:-IsOnCircle(C6, CC), geometry:-draw([CC(color = blue), C1(color = red), C2(color = red), C3(color = red), C4(color = red), C5(color = red), C6(color = red), T(color = black), mA(color = black), mB(color = black), mC(color = black), c1(color = green), c4(color = green), c2(color = green), c3(color = green), c5(color = green), c6(color = green)], symbol = solidcircle, symbolsize = 15, thickness = 2) end proc;

SixPoints(1.5, 1.2);

true, true, true, PLOT(...)

SixPoints(1.5, 1.2)[4];

See geom1.mw

To be continued (The general case will be considered in part 2http://www.mapleprimes.com/posts/200210-Six-Points-On-Circumference-2 .).